Expanding a square

Average Error: 39.0 → 0

Time: 6.3s

Precision: 64

Math

\[\left(x + 1\right) \cdot \left(x + 1\right) - 1\]

↓

\[\mathsf{fma}\left(x, x, x \cdot 2\right)\]

C

double f(double x) {
        double r307308 = x;
        double r307309 = 1.0;
        double r307310 = r307308 + r307309;
        double r307311 = r307310 * r307310;
        double r307312 = r307311 - r307309;
        return r307312;
}

↓

double f(double x) {
        double r307313 = x;
        double r307314 = 2.0;
        double r307315 = r307313 * r307314;
        double r307316 = fma(r307313, r307313, r307315);
        return r307316;
}

Error

Bits error versus x

Derivation

1. Initial program 39.0

   \[\left(x + 1\right) \cdot \left(x + 1\right) - 1\]

2. Simplified 0.0

   \[\leadsto \color{blue}{\left(x + 2\right) \cdot x}\]

3. Taylor expanded around 0 0.0

   \[\leadsto \color{blue}{2 \cdot x + {x}^{2}}\]

4. Simplified 0

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, x \cdot 2\right)}\]

5. Final simplification 0

   \[\leadsto \mathsf{fma}\left(x, x, x \cdot 2\right)\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x)
  :name "Expanding a square"
  (- (* (+ x 1) (+ x 1)) 1))